We propose a method (the ‘gap statistic’) for estimating the number of clusters (groups) in a set of data. The technique uses the output of any clustering algorithm (e.g. K-means or hierarchical), comparing the change in within-cluster dispersion with that expected under an appropriate reference null distribution. Some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature.

Estimating the number of clusters in a data set via the gap statistic

Estimating the number of clusters in a dataset via the gap statistic

For random variables with a unimodal Legesgue density, the 3[sgrave] rule is proved by elementary calculus. It emerges as a special case of the Vysochanskiĭ-Petunin inequality, which in turn is based on the Gauss inequality.

The Three Sigma Rule

In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of Rn, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

/pdf/the-brunn-minkowski-inequality-3ympoqc7tt.pdf

The Brunn-Minkowski inequality

This report summarizes a variety of the most useful and commonly applied methods for obtaining Dempster-Shafer structures, and their mathematical kin probability boxes, from empirical information or theoretical knowledge. The report includes a review of the aggregation methods for handling agreement and conflict when multiple such objects are obtained from different sources.

/pdf/constructing-probability-boxes-and-dempster-shafer-374tnb2noc.pdf

Constructing Probability Boxes and Dempster-Shafer Structures

Properties of Univariate Unimodal Distributions. Concepts of Multivariate Unimodality. Some More Notions of Unimodality. Unimodality for Discrete Distributions. Unimodality of Infinitely Divisible Distributions. Unimodality and Notions of Dependence. Ordering of Distributions by Peakedness. Applications of Unimodality in Statistical Inference. Convexity in Reliability Theory. Appendix.

Unimodality, convexity, and applications

: Document describes The-Gauss-Tchebyshev Inequality for Unimodal Distributions. The determination of a value improves and extends a recent result of Vysochanskii and Petunin who have only considered the case r = 2 with a higher value for k2. The proof is also considerably simpler because it uses the convex structure of the class of unimodal distributions.

The Gauss–Tchebyshev Inequality for Unimodal Distributions

This article presents a large class of probability densities f(x, θ) for which, with positive probability, the maximum likelihood estimator based on a sample of size 2 is non unique, and the possible values of do not form an interval. Such a density f(x, θ) can even be chosen to be unimodal, and one such example is the Cauchy density with a location parameter. A discrete version of the argument gives examples in which the nonuniqueness of the maximum likelihood estimator persists for samples of arbitrary size.

https://www.jstor.org/stable/pdf/2683928.pdf

Examples of Nonunique Maximum Likelihood Estimators

Let X have a multivariate normal distribution. Slepian (1962) proved that the upper and lower orthants (x ^ c) and (x s: ?) have the property that their probabilities are nondecreasing in each p?j. This easily implies, in the bivariate case, that if A = Q\ (J Q3 (J B, where Q\ is an upper quadrant, Qi is a lower quadrant, ? is a disjoint union of horizontal or vertical infinite strips and the interiors ofQ]yQ3 and ? are disjoint, then P(A) is nondecreasing in p. This paper shows that, within a broad class of bivariate regions, sets A of the type described above are the only sets whose probabilities increase with the correlation coefficient when the means and the variances of ?\, X2 take arbitrary values. Some results are also given for the cases where the means and the variances are restricted in some way.

/pdf/regions-whose-probabilities-increase-with-the-correlation-2yuad4jexi.pdf

Regions whose probabilities increase with the correlation coefficient and Slepian's theorem

SYNOPTIC ABSTRACTLet {xn, n ≥ 1} be a sequence of independent identically distributed random variables taking values in Rm and having a common distribution function F. Let Sn = X1, + … + Xn, where addition is coordinatewise. Call F recurrent if the random walk {Sn} is recurrent. Shepp (1962) has used certain definitions of unimodality and peakedness to show that if F and G are symmetric unimodal and F is less peaked than G, then the recurrence of F implies the recurrence of G. This paper extends Shepp's result to a wider class of “symmetric” and “unimodal” distributions.

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA133807&Location=U2&doc=GetTRDoc.pdf

S. W. Dharmadhikari

Papers

Unimodality, convexity, and applications

The Gauss–Tchebyshev Inequality for Unimodal Distributions

Examples of Nonunique Maximum Likelihood Estimators

Regions whose probabilities increase with the correlation coefficient and Slepian's theorem

Recurrence of Symmetric Random Walks